%% LBE macros
\newcommand{\vect}[1]{{ {\bf #1  }}}
\newcommand{\uvect}[1]{{ \hat{\bf #1  }} }
\newcommand{\ci}{
        {
                { {\bf c}}_i
        }
}
\newcommand{\fixme}[1]{

{ \bf{ ***NOTE: #1 }}

}
\newcommand{\half}{\frac{1}{2}}
%collision operator
\newcommand{\cop}{\Omega_i^\sigma}
\newcommand{\copbgk}{{\Omega_i^\sigma}_{\mathrm{BGK}}}
\newcommand{\Dop}{\mathbb D}
% LBGK variables
\newcommand{\tausig}{\tau^\sigma}
\newcommand{\xsig}{x^\sigma}
\newcommand{\tausigb}{\tau^\bar{\sigma}}
\newcommand{\psisig}{\psi^\sigma}
\newcommand{\psisigb}{\psi^{\bar{\sigma}}}
\newcommand{\nusig}{\nu^\sigma}
\newcommand{\msig}{m^\sigma}
\newcommand{\nsig}{n^\sigma}
\newcommand{\usig}{u^\sigma}
\newcommand{\usiga}{u^\sigma_\alpha}
\newcommand{\Fsig}{F^\sigma}
\newcommand{\Fsiga}{F^\sigma_\alpha}
\newcommand{\upr}{u'}
\newcommand{\upra}{{u'}_\alpha}
\newcommand{\vsig}{v^\sigma}
\newcommand{\vsiga}{v^\sigma_\alpha}
\newcommand{\sumsig}{\sum_\sigma}
\newcommand{\sumsigb}{\sum_{\bar{\sigma}}}
\newcommand{\sumsigsigb}{\sum_{\sigma\bar{\sigma}}}
\newcommand{\sumi}{\sum_i}
\newcommand{\msi}{\msig\sum_i}
\newcommand{\ciao}{c_{i{\alpha_1}}}
\newcommand{\cian}{c_{i{\alpha_n}}}
\newcommand{\cia}{c_{i\alpha}}
\newcommand{\cib}{c_{i\beta}}
\newcommand{\cig}{c_{i\gamma}}
\newcommand{\cid}{c_{i\delta}}
\newcommand{\cs}{c_{\mathrm{s}}}
\newcommand{\rhosig}{\rho^\sigma}
\newcommand{\frt}{\frac{\rhosig}{\tausig}}
\newcommand{\xt}{(\vect{x},t)}
\newcommand{\xpct}{(\vect{x}+\ci,t)}
\newcommand{\xpc}{(\vect{x}+\ci)}
\newcommand{\Ua}{U_\alpha}
\newcommand{\fis}{f_i^\sigma}
\newcommand{\fisb}{\bar{f}_i^\sigma}
\newcommand{\Nis}{N_i^\sigma}
\newcommand{\NiU}{N_i^\sigma({\bf U})}
\newcommand{\Niu}{N_i^\sigma({\bf u})}
\newcommand{\Nivs}{N_i^\sigma({\bf v}^\sigma)}
\newcommand{\fone}{f_1(\vect{r},\vect{v},t)}
% Lattice tensors
\newcommand{\Ti}{T_i}
\newcommand{\Ta}{{T}^{(1)}_{\alpha}}
\newcommand{\Tab}{{T}^{(2)}_{\alpha\beta}}
\newcommand{\Tabg}{{T}^{(3)}_{\alpha\beta\gamma}}
% Kronecker deltas
\newcommand{\kronab}{\delta_{\alpha\beta}}
\newcommand{\kronag}{\delta_{\alpha\gamma}}
\newcommand{\kronbg}{\delta_{\beta\gamma}}
% Order zero, one two, etc: to produce f^(0), f^(1) etc.
\newcommand{\oz}[1]{{#1}^{(0)}}
\newcommand{\oo}[1]{{#1}^{(1)}}
\newcommand{\ot}[1]{{#1}^{(2)}}
\newcommand{\ordn}[1]{{#1}^{(n)}}
% Partial derivatives
\newcommand{\partiald}[2]{
        \frac { \partial #1 } { \partial #2 }
}
\newcommand{\partialdd}[2]{
        \frac { \partial^2 #1 } { \partial {#2}^2 }
}
\newcommand{\partialopop}[1]{
        \frac { \partial^2 } { \partial {#1}^2 }
}
\newcommand{\dal}{\partial_\alpha}
\newcommand{\dbe}{\partial_\beta}
\newcommand{\dga}{\partial_\gamma}
\newcommand{\dt}{\partial_{t}}
\newcommand{\dit}{\partial_{1t}}
\newcommand{\dtt}{\partial_{2t}}
% Momentum tensors
\newcommand{\Psa}{\Pi^\sigma_\alpha}
\newcommand{\Psab}{\Pi^\sigma_{\alpha\beta}}
\newcommand{\Psabg}{\Pi^\sigma_{\alpha\beta\gamma}}
\newcommand{\Pa}{\Pi_\alpha}
\newcommand{\Pab}{\Pi_{\alpha\beta}}
\newcommand{\Pabg}{\Pi_{\alpha\beta\gamma}}
\newcommand{\ep}{\epsilon}

